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Properties

Label 2-882-63.16-c1-0-35
Degree $2$
Conductor $882$
Sign $0.995 + 0.0931i$
Analytic cond. $7.04280$
Root an. cond. $2.65382$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (1.09 − 1.34i)3-s + (−0.499 + 0.866i)4-s + 3.18·5-s + (1.71 + 0.272i)6-s − 0.999·8-s + (−0.619 − 2.93i)9-s + (1.59 + 2.75i)10-s + 3.18·11-s + (0.619 + 1.61i)12-s + (−2.85 − 4.93i)13-s + (3.47 − 4.28i)15-s + (−0.5 − 0.866i)16-s + (0.760 + 1.31i)17-s + (2.23 − 2.00i)18-s + (0.641 − 1.11i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.629 − 0.776i)3-s + (−0.249 + 0.433i)4-s + 1.42·5-s + (0.698 + 0.111i)6-s − 0.353·8-s + (−0.206 − 0.978i)9-s + (0.503 + 0.871i)10-s + 0.959·11-s + (0.178 + 0.466i)12-s + (−0.790 − 1.36i)13-s + (0.896 − 1.10i)15-s + (−0.125 − 0.216i)16-s + (0.184 + 0.319i)17-s + (0.526 − 0.472i)18-s + (0.147 − 0.254i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(882\)    =    \(2 \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.995 + 0.0931i$
Analytic conductor: \(7.04280\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{882} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 882,\ (\ :1/2),\ 0.995 + 0.0931i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.84654 - 0.132839i\)
\(L(\frac12)\) \(\approx\) \(2.84654 - 0.132839i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-1.09 + 1.34i)T \)
7 \( 1 \)
good5 \( 1 - 3.18T + 5T^{2} \)
11 \( 1 - 3.18T + 11T^{2} \)
13 \( 1 + (2.85 + 4.93i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.760 - 1.31i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.641 + 1.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 2.23T + 23T^{2} \)
29 \( 1 + (3.54 - 6.13i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.71 - 8.15i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.80 - 4.85i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.41 + 5.91i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.91 + 5.04i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.02 - 1.78i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.562 - 0.974i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.56 - 2.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.48 - 9.49i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 + (-2.48 - 4.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.06 - 3.58i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4.03 + 6.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.112 - 0.195i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.42 - 12.8i)T + (-48.5 - 84.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.832032870424377222068017624417, −9.146392080122039511627017178263, −8.452838576630676083404162524219, −7.32977581941157334858324290660, −6.78029053849040056732479832092, −5.80074045747594573703804900498, −5.18993893585747778876285429393, −3.57886166778538079775157228155, −2.60515980569640017113562494682, −1.35123332733450804489928228884, 1.78828698966792937214516640034, 2.48828204563254151578096868014, 3.81064746550542420797907432895, 4.62514239265376033564564081336, 5.59959806119460646535982743994, 6.46852038744802868379269643520, 7.68455063741365624215902771319, 9.113672828888600354905626149545, 9.471039271977516877638993739137, 9.812290355918444179311839590916

Graph of the $Z$-function along the critical line